Play in the steering wheel is usually the gear box. It also could be in the steering linkage: tierods, center link or idler arm bushings.
While causing drivability problems suspension play; ball joints, control arm bushings, etc. does not usually cause steering wheel play.
Steering gearboxes are supposed to be adjusted off the car. Many experienced techs can do a decent job on the car but its not easy on the 420s as the exhaust is right over the adjustment.
If you haven't already, you should find yourself a good technician to evaluate this and future situations. I wouldn't recommend a tire store for this. My personal recommendation is an independent specializing in MB that does their own alignment.
Owner 24 bay BSC
Bosch Master, ASE master L1
26 years MB technician